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Comparing Market Mechanism Efficiencies
Financial markets have converged on three dominant execution architectures. The first is the lit limit order book (LOB), deployed by major registered exchanges such as NASDAQ and the New York Stock Exchange. In an LOB, all resting orders are publicly visible and matched continuously in strict price--time priority.
The second is the dark pool, operated by alternative trading venues run by large financial institutions. In dark pools, the same continuous double-auction mechanism as in the LOB applies. However, in dark pools, the order book data is withheld from participants before execution. Only executed orders are publicly disclosed.
The third is the periodic batch auction, in which orders accumulate over a fixed interval and clear simultaneously at a uniform price. The periodic batch auctions eliminate time priority entirely.
All three architectures attract substantial order flow, coexist in live markets, and are subject to active regulatory debate on both sides of the Atlantic. Yet the literature on market microstructure has evaluated these mechanisms almost exclusively through the lenses of liquidity and price discovery: bid--ask spreads, depth, and the informational efficiency of prices. These metrics matter, but they are incomplete. A market that produces tight spreads may still be inefficient in a deeper sense if traders unnecessarily expend significant resources competing for queue position or timing their submissions strategically. Such costs may include traders' time, effort, and foregone outside options. These costs can be real welfare losses that do not appear in conventional liquidity statistics.
This paper asks a different question: when the full cost of participating in a market is taken into account, which of the three architectures produces the highest aggregate welfare? The costs in question include waiting costs, transaction costs, and the social waste generated by strategic timing games.
\paragraph{Main result.} Our central theorem establishes that, under moderate arrival rates and bounded adverse selection, dark pools dominate both lit exchanges and batch auctions in aggregate ex-ante welfare. Formally, we prove: \[ W^{\mathrm{DARK}} \;>\; W^{\mathrm{LIT}} \;>\; W^{\mathrm{BATCH}}. \] The welfare advantage of dark pools over lit exchanges arises not from better prices, but from the elimination of strategic timing games. We argue that prices are transfers and do not affect total surplus. Furthermore, when traders can observe the order book, they delay or rush submissions to optimize their queue position, generating socially wasteful waiting costs. Market opacity removes this incentive while preserving the ability of urgent traders (those with high waiting costs) to execute immediately via market orders.
The welfare advantage of lit exchanges over batch auctions arises because batch auctions impose mandatory waiting on all traders, including the most time-sensitive, and introduce execution uncertainty through pro-rata rationing.
\paragraph{Why welfare, and why now.} The welfare perspective is particularly timely. In the United States, the Securities and Exchange Commission's 2023 equity market structure reform proposals explicitly revisited the rules governing dark pools and the order competition requirement. The proposals reopened questions about the social value of opacity SEC2023. In Europe, the European Securities and Markets Authority (ESMA) published a landmark Call for Evidence in April 2026 documenting a measurable shift away from lit continuous trading toward closing auctions, frequent batch auctions (FBAs), and systematic internalizers over the 2022--2025 period, and explicitly soliciting views on whether regulatory reform is needed to restore efficient price formation ESMA2026CfE. Both debates hinge, at their core, on the same trade-off this paper formalizes: transparency enables strategic gaming; opacity prevents it but introduces adverse selection. A rigorous welfare comparison of the three architectures is therefore both theoretically overdue and practically urgent.
\paragraph{Modeling approach.} We develop a unified game-theoretic framework that represents all three mechanisms as queueing systems in the tradition of contStoikovTalreha2010. A continuum of risk-neutral traders with heterogeneous private valuations $V_i$ and waiting costs $C_i$ arrive according to a Poisson process and choose among market orders, limit orders, and an outside option. The three mechanisms differ solely in the information available to arriving traders and in the service discipline applied to queued orders:
We characterize Bayesian equilibria under each mechanism (Propositions 1--3), derive aggregate ex-ante welfare for each (Section (ref)), and establish the welfare ranking under explicit sufficient conditions (Theorem (ref)).
\paragraph{Relation to the queueing literature.} Our framework connects market microstructure to two foundational results in queueing theory. Leshno2022 establishes that, in a dynamic matching model with overloaded waiting lists, SIRO is ex-ante welfare-superior to FCFS because it discourages socially excessive effort to secure priority. Applied directly, this result would suggest batch auctions dominate lit exchanges. CheTercieux2023 qualify this finding crucially: when agents are uninformed about their queue position, FCFS is welfare-superior to SIRO because the absence of positional information eliminates the distortions that make SIRO attractive in the first place. Applied to market design, an uninformed FCFS queue---precisely the dark pool---dominates both informed FCFS (lit exchange) and SIRO (batch auction).
Our contribution is to embed these abstract queueing results in a two-sided financial market with endogenous participation, heterogeneous trader types, adverse selection, and an explicit outside option. This embedding is non-trivial: in a financial market, the "service rate" $\lambda(p, B, s)$ depends endogenously on the limit price chosen by the liquidity provider, the book state, and the direction of trade. The equilibrium strategies derived in Section (ref) characterize how traders optimally choose between market orders, limit orders, and exit in each mechanism, and how these choices aggregate to determine social welfare. Section (ref) compares queueing literature in greater detail.
\paragraph{Relation to the market microstructure literature.} The literature on dark pools is extensive. Zhu2013 shows that dark pools attract relatively more uninformed order flow, improving price discovery in lit markets but concentrating adverse selection in the dark. Our model complements this by focusing on the welfare channel that Zhu's model abstracts away from: the strategic timing cost imposed on informed traders by the transparency of the lit book. OHara1998 and Aldridge2013 provide the canonical treatments of the continuous limit-order-book mechanism on which our lit-exchange model is based. BudishEtAl2015 develop the batch auction proposal and the welfare case against continuous-time priority. Our analysis shows that their proposal eliminates one source of welfare loss (the arms race for speed) while introducing two others (forced waiting and execution uncertainty). Our analysis also characterizes the conditions under which the net effect is negative.
Most closely related to our contribution is the work of JohnEtAl2025, who analyze the efficiency of blockchain market microstructure from the perspective of liquidity providers, and CaoEtAl2025, who document that liquidity providers on Uniswap lose money on average. Neither paper considers aggregate welfare across all market participants---the perspective we adopt here. To our knowledge, this paper is the first to derive a complete welfare ranking of lit exchanges, dark pools, and batch auctions from first principles in a unified model.
\paragraph{Key mechanisms and intuition.} The welfare ranking $W^{\mathrm{DARK}} > W^{\mathrm{LIT}}$ rests on a single insight: observable queue position creates negative externalities. A trader who observes a thin book in a lit market may strategically wait for conditions to improve before submitting, incurring a waiting cost with no social benefit. A trader who observes a thick book may rush to capture time priority, submitting at a worse price than they would otherwise accept. In aggregate, these reactions to the public order book raise total waiting costs above the socially optimal level. The dark pool removes the informational input that drives these responses. High-cost traders still execute immediately via market orders (their waiting cost $C_i$ exceeds the cutoff $\bar{C}^*$); low-cost traders still post limit orders; but neither group conditions these choices on the unobservable book state, $B_t$. The resulting adverse selection affects the distribution of surplus between buyers and sellers, but not the total surplus, because prices are transfers.
The ranking $W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$ rests on two separate inefficiencies introduced by batch clearing. First, mandatory waiting: every trader who arrives in the interval $[(k-1)T, kT)$ must wait until $kT$ to execute, even those with arbitrarily high waiting costs who would ordinarily submit market orders for immediate execution. This generates an aggregate welfare loss of $\mathbb{E}[C] \cdot (T/2) \cdot N$, which is strictly positive for any batch interval $T > 0$. Second, execution uncertainty: when orders on one side of the market exceed those on the other at the clearing price, pro-rata rationing leaves some traders unfilled. The prospect of non-execution reduces the expected surplus from participation, causing some traders whose participation would be socially efficient to take their outside option instead, generating deadweight loss. A lit market with a continuous auction eliminates both inefficiencies for urgent traders: market orders execute immediately and with certainty.
\paragraph{Sufficient conditions and robustness.} The welfare ranking is not unconditional. Theorem (ref) establishes sufficient conditions: arrival rate $\lambda$ in a moderate range $[\lambda_{\min}, \lambda_{\max}]$, valuation dispersion $\Delta < \Delta_{\max}$ (bounded adverse selection), and reporting delay $\delta < T/2$ in the dark pool. The arrival-rate condition ensures that the dark pool market is thick enough that high-cost traders can rely on near-immediate execution via market orders; if $\lambda$ is very low, high-cost traders may face excessive wait times even in the dark pool, potentially reversing the ranking with the lit exchange. Traders' inability to condition on $B_t$ causes a pricing distortion in the dark pool. The bounded adverse selection condition ensures that the pricing distortion does not become so severe that it materially reduces participation below the lit-exchange level. Section (ref) characterizes these boundary conditions and discusses when they are likely to hold in practice.
\paragraph{Regulatory implications.} Our results speak directly to ongoing regulatory debates. First, proposals to require dark pools to disclose pre-trade order book information (so-called pre-trade transparency mandates) would eliminate the information design that generates their welfare advantage. Our model predicts that such rules would reduce aggregate welfare if the strategic timing costs in lit markets are significant. Second, ESMA's current review of periodic auction mechanisms in European equity markets ESMA2026CfE raises the question of whether FBAs should be subject to tighter transparency or priority rules. Our model suggests that the welfare cost of FBAs relative to continuous markets is primarily the forced-waiting loss, not the opacity; regulatory interventions that shorten the batch interval $T$ would reduce this cost, while transparency requirements would not. Third, the finding that dark pools' welfare advantage is conditional on liquid lit markets providing reference prices (Section (ref)) supports a policy of maintaining a mandatory minimum share of lit trading, as several regulators have considered. This would prevent the free-riding dynamic from destabilizing the price discovery infrastructure on which dark pools depend.
\paragraph{Organization.} The remainder of the paper is structured as follows. Section (ref) represents each mechanism as a queueing system and provides background. Section (ref) defines the model environment, trader types, and utility specification. Section (ref) characterizes the Bayesian equilibrium under each mechanism. Section (ref) defines the welfare measure and proves the main welfare ranking (Theorem (ref)). Section (ref) presents the results of numerical simulations. Section (ref) considers the robustness of the analysis. Section (ref) discusses dynamic considerations, robustness, and regulatory implications. Section (ref) concludes. Proofs omitted from the main text appear in the Appendix.
This paper's welfare ranking draws on two foundational results in the theory of queue design: Leshno2022 and CheTercieux2023. The existing literature on dark pools and batch auctions has not connected these results to market microstructure. We do so here, but the connection requires care: both papers prove things that differ from what this paper needs, and the differences matter for what is novel in our contribution.
\paragraph{Leshno2022.} Leshno2022 studies a single-sided waiting list in which heterogeneous agents choose between items that arrive stochastically over time, each associated with an endogenously determined expected wait. The central finding is that waiting-time fluctuations lead to misallocation and welfare loss, and that a simple SIRO randomized assignment policy can reduce these fluctuations and thereby increase welfare relative to FCFS. The mechanism is allocation efficiency under item heterogeneity: when wait times are dispersed across states, some agents accept mismatched items to avoid long waits, causing allocative inefficiency. SIRO reduces variation in wait times across queue states, maintaining an acceptable expected wait under a larger range of states than FCFS, and is characterized as the robustly optimal mechanism.
Three features of Leshno's model are essential for his result. First, agents choose between heterogeneous items (e.g., a preferred organ type versus a mismatched one); the welfare gain from SIRO comes from reducing preference mismatches, not from reducing aggregate waiting costs. Second, the queue is one-sided: agents wait for items that arrive exogenously; there is no strategic order placement or pricing. Third, the model is overloaded by assumption: demand permanently exceeds supply, so the designer wishes to incentivize agents to queue as long as possible. SIRO achieves this by smoothing out wait-time fluctuations that would otherwise induce premature exits.
\paragraph{CheTercieux2023.} CheTercieux2023 study a mechanism design problem in which a queue designer jointly chooses the service discipline and the information disclosed to agents. The optimal mechanism has a cutoff structure: agents are induced to enter up to a certain queue length and never to exit; they are served according to FCFS; and they are given no information throughout the process beyond the designer's recommendations.
The mechanism behind this result is belief regulation. Under FCFS, the passage of time in the queue is good news for a waiting agent, because surviving in the queue signals that fewer agents remain ahead. This makes the incentive to stay in the queue self-reinforcing. Under SIRO and other rules with dispersed wait times, the elapse of time without being served signals a longer residual wait, undermining the incentive to remain. This is the fundamental issue that CheTercieux2023 identify. Withholding positional information (no information beyond the designer's recommendations) ensures that agents form beliefs consistent with the FCFS discipline's favorable belief dynamics, eliminating the incentive to exit prematurely.
Two features of their model are essential. First, the welfare criterion combines agent utility and the service provider's payoff; FCFS is optimal partly because it maximizes service utilization, which benefits the provider. Second, the result is about the joint optimization of discipline and information. The key finding is not that FCFS dominates SIRO unconditionally, but that FCFS combined with no positional information dominates SIRO (or any other rule) combined with any information structure. The information-design component is essential; FCFS with full information is not generally optimal.
If Leshno's result applied directly here, it would imply $W^{\mathrm{BATCH}} > W^{\mathrm{LIT}}$, because SIRO (batch auctions) would dominate FCFS (lit exchanges). Our main result finds the opposite. If Che and Tercieux's result applied directly, it would imply that the optimal mechanism is FCFS with no information, which corresponds to the dark pool — but their result applies to a single-sided designer-controlled queue, not a two-sided competitive market with endogenous order placement and an outside option. Four structural differences explain why direct application fails in both cases.
\paragraph{Difference 1: One-sided versus two-sided.} Both queueing papers study single-sided queues in which one population (agents) waits for a second population (items or servers) whose arrival is exogenous. A financial market is fundamentally two-sided: buyers and sellers both choose endogenously whether to enter, which order type to submit, and at what price. In a two-sided market, a trader can bypass the queue entirely by submitting a market order — paying the spread to receive immediate execution. This option, absent in any single-sided queue model, changes the welfare calculus decisively.
Under FCFS in a financial market, high-cost traders (large $C_i$) can avoid waiting altogether by using market orders, incurring zero waiting costs. The Leshno model has no such escape valve, which is why SIRO appears beneficial there: it smooths wait-time fluctuations for agents who must wait. In our model, those same high-cost traders never wait — so the smoothing benefit of SIRO is irrelevant, and its cost dominates. The cost is the forced $T/2$ wait for all traders, including the impatient.
\paragraph{Difference 2: Endogenous versus exogenous arrival.} In Leshno's model, the flow of agents joining the pool is exogenous. This means that maximizing welfare is equivalent to maximizing allocative efficiency, and the trade-off between allocative efficiency and congestion that arises with endogenous arrival is absent.
In our model, participation is endogenous: traders compare the expected surplus from entering the market against the outside option (equation 2). The participation margin is central to all three welfare comparisons. For the $W^{\mathrm{DARK}} > W^{\mathrm{LIT}}$ result, the outside option determines which types are excluded by adverse selection. For $W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$, the execution uncertainty in batch auctions causes marginal traders to exit, generating deadweight loss (Remark 2). Neither of these effects exists in a model with exogenous arrival.
\paragraph{Difference 3: Item heterogeneity versus price optimization.} Leshno's welfare gain from SIRO comes from reducing preference mismatches: agents are assigned mismatched items under FCFS because dispersed wait times lead them to accept a worse item rather than waiting. In a financial market, there is a single asset, and all trades are at market-clearing prices; there is no item heterogeneity and no mismatch in Leshno's sense. The relevant welfare source is the price-quality trade-off in limit-order placement: a trader chooses a limit price to balance execution speed against price improvement, and the social cost arises from the strategic timing of this choice. This is structurally different from the mismatch problem, and SIRO's solution to mismatch (smoothing wait-time dispersion) does not address the strategic timing problem.
\paragraph{Difference 4: Designer control versus competitive equilibrium.} Che and Tercieux assume a single designer who controls both the queuing discipline and the information disclosed to agents and who can commit to recommendations. The dark pool in our model is not a mechanism designed by a welfare-maximizing planner. Instead, it is a market institution whose opacity arises from proprietary considerations and regulatory permissions and whose equilibrium is determined by competitive trader behavior. The Che--Tercieux result establishes that a planner would choose FCFS-with-no-information. A dark pool is approximately an FCFS-with-no-information. Our result, therefore, establishes that the dark pool generates higher welfare than observable-book alternatives in the competitive equilibrium. This is not a contradiction but a reinforcement: the planner's choice in their model is precisely the institution our paper identifies as welfare-superior in ours.
Given these differences, what does the present paper contribute beyond combining and applying Leshno2022 and CheTercieux2023? We identify five contributions that are not present in either paper.
\paragraph{Contribution 1: The market-order escape valve.} The most important structural novelty is the option to submit a market order. In a two-sided financial market, high-cost traders can always pay the spread for immediate execution. This option transforms the welfare comparison: SIRO's benefit (smoothing wait times for traders who must wait) is dominated by its cost (forcing all traders to wait, including those who would otherwise use market orders). Formally, the aggregate welfare loss from batch auctions is $\Delta W_{\mathrm{wait}} = \mathbb{E}[C] \cdot (T/2) \cdot N > 0$ for any $T > 0$ and any $\mathbb{E}[C] > 0$ (Remark 1). This term has no counterpart in the Leshno model because agents cannot bypass the queue. Its existence explains why $W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$ even in parameter regions where Leshno's logic would suggest SIRO dominates.
\paragraph{Contribution 2: Endogenous participation and deadweight loss.} With endogenous participation, execution uncertainty in batch auctions generates deadweight loss (Remark 2, equation 59): some traders who would participate under a continuous mechanism exit when the pro-rata fill rate makes expected utility fall below the outside option. This deadweight loss is absent from both reference papers. It provides an additional welfare disadvantage of SIRO beyond the forced-waiting cost. The magnitude of the deadweight loss depends on the distribution of trader types and the outside-option value; these are parameters that can be calibrated empirically.
\paragraph{Contribution 3: Adverse selection under FCFS-with-opacity.} CheTercieux2023 establish that no-information FCFS is optimal for a planner, but do not characterize the adverse selection that opacity creates when traders have heterogeneous private information. In a financial market, dark-pool traders execute at prices that may differ from the true fundamental value $\hat{v}$ because they cannot condition on the book state $B_t$. Section (ref) of this paper characterizes the two channels through which adverse selection affects welfare (the price channel and the participation channel), establishes when each is dominated by the timing-game saving, and derives the threshold $\Delta^*(\lambda)$ above which adverse selection reverses the welfare ranking. This characterization is novel and has no counterpart in either reference paper.
\paragraph{Contribution 4: Competitive equilibrium, not mechanism design.} This paper characterizes welfare in the competitive Bayesian equilibria of three market mechanisms (Propositions 1--3), not in a planner-designed mechanism. The equilibrium concept is Bayes--Nash rather than dominant-strategy incentive compatibility. Market orders, limit orders, and outside options are all available, and traders choose among them strategically. The equilibrium strategies (cutoff functions $C^*(V,B)$, $\bar{C}^*$, and $\tau^*(V,C)$) are derived endogenously, not imposed by a designer. This is the appropriate concept for evaluating real market institutions, and it generates predictions about observable behavior (e.g., participation rates, order-type composition, and welfare gaps) that the mechanism-design approach does not.
\paragraph{Contribution 5: A complete ranking with sufficient conditions.} Both reference papers compare two mechanisms (SIRO vs.\ FCFS, or informed vs.\ uninformed FCFS). This paper compares three — dark pool, lit exchange, and batch auction — and derives a complete ranking $W^{\mathrm{DARK}} > W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$ with explicit sufficient conditions on the arrival rate $\lambda$, valuation dispersion $\Delta$, and reporting delay $\delta$. Section (ref) characterizes these conditions precisely and identifies the parameter regions in which each inequality reverses. This complete, conditional ranking is new to the literature on market mechanism comparison.
Table (ref) summarizes the key differences between this paper and the two reference papers across five modeling dimensions.
\addcontentsline{toc}{section}{Notation}
The following table defines every symbol used in the paper in the order it first appears. Symbols are grouped by conceptual category. Where a symbol is overloaded or has a mechanism-specific variant, the variants are listed together. Equation and section numbers in the rightmost column give the location of the formal definition or first substantive use.
Disambiguation notes. Three symbols require explicit disambiguation because they are used with similar (or identical) typography for conceptually distinct objects.
Most financial market mechanisms can be represented as queueing systems.
As shown in contStoikovTalreha2010, a market mechanism for traditional registered exchanges can be represented as a First Come First Served queuing system (FCFS). This model is formally known as a double-sided continuous auction. In this model, all resting (limit) orders to buy and sell are arranged by price and, within each price "queue", their arrival sequence. Limit orders are analogous to waiting customers in the traditional queuing literature. The arriving market orders are matched with the best-priced earliest-arrival limit orders. The market orders are the "servers" that complete an FCFS mechanism. The FCFS markets have not been immune to manipulation. For example, CongEtAl2022 documents issues in the Bitcoin markets.
Dark pools are also double-sided continuous auctions, just like lit exchanges, with one important informational difference. While the exchanges distribute various details about the composition of their respective queuing systems, known as limit order books (LOBs), dark pools keep their LOBs hidden from all market participants. However, by law, even dark pools are required to make public all trade details after each trade (limit- and market order matching) occurs. Dark pools are also not perfect. For example, Zhu2013 explored manipulation issues in dark pools.
Batch auctions, as a theoretical market-design mechanism, were proposed by BudishEtAl2015 as a response to the welfare costs of high-frequency trading arms races in continuous markets. In the batch auction setup, a mini-auction takes place at fixed time intervals, and all orders submitted within each interval are executed simultaneously at a uniform clearing price determined by aggregate supply and demand. BudishEtAl2015 argued that eliminating continuous-time priority competition reduces socially wasteful investment in speed and narrows bid--ask spreads. However, as this paper shows, batch auctions introduce distinct welfare costs of their own, most notably through forced waiting and execution uncertainty.
As a queueing system, each batch auction pools orders for a fixed window and then clears them collectively. Within each batch, there is no time priority: orders are matched in Service in Random Order (SIRO), meaning all orders at the clearing price have an equal probability of execution, regardless of submission timing. This is the defining structural feature we analyze formally in Section (ref).
\paragraph{The EU as the relevant real-world context.} The most policy-relevant deployment of periodic batch auction mechanisms is not in blockchain settings but in European equity markets, where Frequent Batch Auctions (FBAs) have emerged as a distinct and growing trading-venue category under the MiFID II/MiFIR regulatory framework. Since MiFID II came into application in January 2018, FBAs rapidly gained market share by offering an alternative to both lit continuous order books and dark pools. Unlike continuous auctions, FBAs run repeated short-duration auctions throughout the trading day, with orders matched at a uniform clearing price determined within the prevailing best bid--offer spread.
The EU regulatory debate around FBAs is directly relevant to the welfare questions this paper addresses. ESMA launched its first Call for Evidence on FBAs in November 2018, specifically because of concerns that they were being used to circumvent the MiFID II Double Volume Cap (DVC)---a cap designed to limit dark trading by restricting how much volume could execute under pre-trade transparency waivers ESMA2018CfE. ESMA identified four main characteristics of FBA systems warranting regulatory scrutiny: limited pre-trade transparency, short auction duration, price determination within the best bid--offer price, and self-matching features ESMA2019FinalReport. Following its assessment, ESMA published an Opinion in October 2019 clarifying that trading venues operating FBA systems must inform market participants that an auction has started, thereby enabling genuine pre-trade transparency. The Opinion also set out several practices that may undermine the price formation process ESMA2019Opinion.
As of the writing of this paper, the debate is active and escalating. ESMA launched a new Call for Evidence in April 2026 documenting a measurable decline in lit continuous trading between 2022 and 2025, offset by increased activity in closing auctions, frequent batch auctions, and systematic internalizer (SI) trading ESMA2026CfE. The 2026 Call for Evidence provides a granular analysis of periodic auctions specifically, and solicits stakeholder views on whether the current market structure delivers efficient price formation, transparency, and execution outcomes, and where regulatory adjustments may be needed. ESMA expects to publish a feedback statement in Q3 2026.
This paper's welfare framework offers a theoretical foundation for several of the open questions ESMA is now asking empirically. We highlight three direct connections.
\paragraph{Why the EU context is theoretically appropriate.} The BudishEtAl2015 batch auction proposal was designed explicitly to maximize social welfare by eliminating the arms race for speed. European FBAs are deployed within a regulatory framework, MiFID II, that shares this welfare and transparency orientation, making the EU setting a coherent context in which to evaluate our model's predictions. Critically, European FBAs implement a genuine uniform clearing-price auction with SIRO execution within each batch, which is the mechanism our formal model in Section (ref) analyzes. This institutional alignment makes quantitative and qualitative comparison between the theory and the data possible in principle.
\paragraph{Scope of the batch auction model.} This paper analyzes the batch auction mechanism in its canonical BudishEtAl2015 form: fixed-interval, uniform clearing price, SIRO within each batch. Real-world FBAs vary in their auction trigger mechanisms (some are time-triggered, while others are order-triggered) and in their price determination rules (some require a reference price from the lit market). These design variations affect the quantitative magnitude of the welfare effects derived in Section (ref), but not the qualitative ranking, provided the fundamental SIRO structure and the forced-waiting property are preserved. Future work could extend the model to study trigger-based auction designs, which introduce an additional layer of strategic order timing that our current framework abstracts away from.
Next, each of the market queueing models is analyzed in detail.
Consider a continuous-time trading environment with time $t\in[0,\infty)$. There is a single risky asset with fundamental value $v \sim \mathcal{N}(\hat{v}, \sigma_v^2)$, where $\hat{v}$ is commonly known, but $\sigma_v^2$ represents residual uncertainty. A mass-1 continuum of risk-neutral traders arrives according to a Poisson process with intensity $\lambda$.
Each trader $i$ is characterized by a type $\theta_i = (V_i, C_i, s_i, \tau_i)$, where $V_i$ represents the private valuation benefit, $C_i$ is the cost per unit of waiting time, $s_i\in\{-1,+1\}$ is the desired trade direction (buy or sell) and $\tau_i$ is the arrival time. We assume that the types are drawn independently from a joint distribution F(V,C,s) with continuous density f.
Each trader's utility depends on whether and when they trade, the execution price, and the waiting costs incurred. Formally, trader $i$ who executes at time $t + w_i$ (where $w_i \geq 0$ is waiting time) at price $p$ obtains utility:
where $K > 0$ is a fixed transaction cost (including exchange fees, market impact, etc.). If the trader $i$ does not execute, he receives the utility of the outside option:
where $\gamma \in[0,1]$ represents the extent to which private value can be realized outside the market (e.g., through delayed trading or alternative venues), and $K_o\geq 0$ is the cost of pursuing the outside option. We assume that $K_o<K$ makes participation in the market valuable.
The objective of this note is to consider whether one of the three market models results in higher aggregate welfare for all agents in a given system. As shown by Leshno2019, the Service In Random Order (SIRO) policy deployed by batch auctions is superior in terms of total welfare to the traditional exchanges' First Come First Served (FCFS) design. As shown by Leshno2019, SIRO ex-ante maximizes the aggregate welfare of both customers and servers.
According to CheTercieux2023, however, in the absence of information, the First Come First Served (FCFS) policy provides an even more efficient outcome than SIRO. Specifically, being unaware of one's position in the queue produces ex-ante welfare-efficient outcome, while preserving other desirable properties, such as the fairness of the process and incentive compatibility. This implies that dark pools are the preferred market design among the three models considered in this note.
The lit exchange operates as a continuous double auction with a transparent limit order book (LOB). At any time $t$, the state of the market is summarized by the order book state $B_t = {(p_j, q_j, \tau_j)}$, where $p_j$ is the limit price, $q_j$ is quantity, and $\tau_j$ is submission time. $B_t$ is publicly observable.
Order execution follows First-Come-First-Served (FCFS) priority within each price level. The matching rule is as follows: a market buy order of size $q$ is matched with the lowest-priced limit sell orders, in order of their submission time, until quantity $q$ is exhausted (or vise versa for market sell orders).
Strategic considerations: Traders observe $B_t$ and optimally choose between submitting a market order (immediate execution at the best available price), a limit order (joining the queue at the chosen price), or waiting. The transparency of $B_t$ creates strategic complementarities in order submission timing.
Dark pools operate identically to lit exchanges in terms of the matching mechanism (FCFS continuous auction), but with a critical informational difference: the order book state $B_t$ is unobservable to all traders before execution. Traders know only the distribution of order flow, not the realized state.
However, dark pools are transparent in trade results: after each execution, the trade (price, quantity, timestamp) is publicly reported with delay $\delta > 0$. This creates a filtered information structure in which traders can infer some properties of $B_t$ from past trades, but cannot observe the current state.
The opacity of $B_t$ eliminates strategic timing games based on queue position, but introduces adverse selection risk. Traders must form beliefs about $B_t$ using only distributional information and delayed trade reports.
Periodic batch auctions are clear at discrete intervals $T, 2T, 3T,\dots$ with period $T > 0$. All orders submitted during the interval $[(k-1)T, kT)$ are batched and executed simultaneously in time $kT$ at a uniform clearing price $p_k$ that maximizes the executed volume.
Within each batch, orders are matched in Service-In-Random-Order (SIRO), i.e., all orders at the clearing price have equal probability of execution regardless of submission timing. If demand exceeds supply at $p_k$ (or vice versa), the rationing is uniform random.
SIRO eliminates time priority, so there is no benefit in being the first within a batch. However, traders still strategically choose which batch to enter and face uncertain execution due to pro-rata allocation when orders are unbalanced.
We formally define the information available to trader $i$ arriving at time $\tau_i$ under each mechanism:
In dark pools and batch auctions, traders cannot observe the current order book state and must form beliefs. Let $\beta_i(B_{\tau_i} | I_i)$ denote the belief of the trader $i$' about the current order book state conditional on their information set.
Under rational expectations equilibrium, beliefs are consistent with the equilibrium distribution of order flows. For dark pools, traders use Bayes' rule to update beliefs based on delayed trade reports:
where $L(\cdot|\cdot)$ is the likelihood of observing the delayed trading history given the current state of the order book. In equilibrium, these beliefs must be correct on average.
In the lit exchange, traders observe the full order book and optimally choose their order submission strategy. Define the value function for a trader of type $\theta = (V,C,s,\tau)$ who observes the book state $B$:
where $V_{market}$ is the value of submitting a market order, $V_{limit}$ is from a limit order, $V_{wait}$ is waiting, and $U_o$ is the outside option. The optimal strategy $\sigma^{LIT}(\theta,B)$ selects the action that reaches the maximum.
Under Assumption (ref), we can take the derivative of $U_{\text{LO}}$ with respect to $p$. For notational simplicity, consider a buy order ($s = +1$); the sell order case follows by symmetry. The first-order condition is:
Using the chain rule:
Substituting:
Rearranging gives the implicit equation for $p^*$:
This implicitly defines $p^*(V, C, s, B)$ as the solution to equation (ref) for given $(V, C, s, B)$. Note that $V$ does not appear in equation (ref), so $p^*$ depends on $(C, s, B)$ but not on $V$ in the first-order condition. However, $V$ affects the participation decision through the level of $U_{\text{LO}}$.
Dark pool traders cannot observe B and must optimize based on beliefs. The value function becomes:
where expectations are taken over beliefs $\beta(B|I)$. The key difference is that traders do not strategically time their orders based on the observed queue position, as this information is not available.
The stark contrast between Proposition (ref) (dark pools) and the lit exchange equilibrium (where cutoffs depend on $B$) highlights the fundamental role of the information structure.
In batch auctions, traders arriving in interval $[(k-1)T, kT)$ choose whether to enter the current batch, wait for the next batch, or exit. Within a batch, SIRO eliminates timing advantages, but execution is uncertain due to pro-rata rationing.
The batch auction structure creates two distinct sources of inefficiency relative to continuous markets:
We define aggregate ex-ante welfare as the expected total surplus across all traders before types are realized:
where $M \in \{LIT, DARK, BATCH\}$ denotes the mechanism, $\sigma^M$ is the equilibrium strategy profile, and expectations are taken over all uncertainties (fundamental value, matching outcomes, etc.). This welfare measure includes waiting costs and transaction costs, but excludes the transfers between buyers and sellers (prices), which are pure redistribution. It captures efficiency in the sense of the total surplus generated by the matching process.
Our central results compare welfare across the three mechanisms. The key insight is that information opacity in dark pools eliminates socially wasteful strategic behavior, despite creating adverse selection.
The theoretical results of Section (ref) establish the welfare ranking $W^{\mathrm{DARK}} > W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$ under the sufficient conditions of Theorem (ref): a moderate arrival rate $\lambda \in [\lambda_{\min}, \lambda_{\max}]$, bounded valuation dispersion $\Delta < \Delta_{\max}$, and a short reporting delay $\delta < T/2$. This section complements those results with a calibrated numerical simulation that serves three purposes. First, it verifies that the welfare ranking is quantitatively robust (and not merely an artifact of limiting arguments) at parameter values representative of actual equity markets. Second, it maps the boundary of the parameter region in which the ranking holds, directly addressing the robustness concern raised in the literature on mechanism comparison BudishEtAl2015, Zhu2013. Third, it produces four figures that illustrate the distinct economic mechanisms driving each inequality, giving the theoretical results an empirical interpretation.
The simulation draws $n = 20{,}000$ traders per experiment from the joint type distribution $F(V, C, s)$ specified in Assumption 1. Private valuations are drawn from $\text{Uniform}[\hat{v} - \Delta,\, \hat{v} + \Delta]$ with $\hat{v} = 100$ and baseline $\Delta = 2$, corresponding to a valuation half-spread of two percent of the asset's fundamental value. This is broadly consistent with the dispersion of private information documented in equity-market studies OHara1998. Waiting costs are drawn from $\text{Exp}(\mu_C)$ with $\mu_C = 2$, giving $\mathbb{E}[C] = 0.5$. The arrival rate is set to $\lambda = 5$ at baseline. The fixed transaction cost is $K = 0.05$ and the outside-option cost is $K_o = 0.02 < K$, satisfying the participation constraint of Assumption 2. The batch interval is $T = 1$ and the dark-pool reporting delay is $\delta = 0.1 < T/2 = 0.5$, satisfying the condition of Theorem (ref).
At these baseline parameters, the simulation yields the welfare estimates reported in Table (ref).
The full ranking $W^{\mathrm{DARK}} > W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$ holds at baseline with comfortable margins: the dark pool generates welfare $17.0\%$ above the lit exchange, and the lit exchange generates welfare $51.7\%$ above the batch auction. The participation rate ordering mirrors the welfare ordering, with the batch auction exhibiting the lowest rate (39.9%) and the dark pool the highest (46.1%), consistent with Remarks 1 and 2.
Figure (ref) plots aggregate welfare $W^M$ for each mechanism as $\lambda$ varies from 1 to 12, holding all other parameters at their baseline values. The right panel plots the two pairwise welfare gaps, $W^{\mathrm{DARK}} - W^{\mathrm{LIT}}$ and $W^{\mathrm{LIT}} - W^{\mathrm{BATCH}}$, with the shaded band indicating the Theorem (ref) region $[\lambda_{\min}, \lambda_{\max}]$.
Several features of Figure (ref) warrant discussion.
\paragraph{All three mechanisms benefit from thicker markets.} Welfare is increasing in $\lambda$ across all three mechanisms. This reflects the gains-from-trade effect: a higher arrival rate raises the probability that a buyer and seller with compatible valuations meet within any given time window, increasing the expected surplus from participation. The effect is monotone and concave, consistent with the diminishing returns to market thickness documented in the matching literature Leshno2022.
\paragraph{The dark pool's advantage is largest at low arrival rates.} The gap $W^{\mathrm{DARK}} - W^{\mathrm{LIT}}$ is largest near $\lambda = 1$ (approximately 0.095) and declines monotonically as $\lambda$ increases, falling to approximately 0.022 at $\lambda = 12$. This pattern has a clear economic interpretation. When markets are thin, the order book state $B_t$ is highly variable and informative: a trader who observes a thin book at a particular moment gains substantial information about near-term execution prospects, and the incentive to time submissions strategically is therefore strong. As $\lambda$ increases, the book fills rapidly, and its state becomes less variable; the information content of observing $B_t$ diminishes, and so does the strategic timing waste it generates. At very high arrival rates, the book is almost always deep, and the cost of observing it is nearly nil, explaining why the two continuous-auction mechanisms converge as $\lambda \to \infty$. Importantly, however, the gap $W^{\mathrm{DARK}} - W^{\mathrm{LIT}}$ remains strictly positive throughout the plotted range, consistent with the theoretical prediction.
\paragraph{The lit exchange's advantage over batch auctions grows with market thickness.} In contrast to the dark-pool gap, $W^{\mathrm{LIT}} - W^{\mathrm{BATCH}}$ is smallest near $\lambda = 1$ (where it is, in fact, briefly negative, indicating that the batch auction marginally dominates the lit exchange in very thin markets) and increases monotonically with $\lambda$, reaching approximately 0.11 at $\lambda = 12$. The negative gap at very low $\lambda$ reflects the fact that in a thin lit market, market orders rarely find counterparties quickly, so the forced-waiting cost of the batch auction (which aggregates orders and thereby increases match probability) is offset by the matching benefit. As $\lambda$ grows, immediate market orders in the lit exchange become reliable, so the probability of finding a counterparty quickly approaches one. As a result, the forced $T/2$ wait in the batch auction becomes a pure cost with no compensating benefit. The lit-exchange advantage therefore increases with market thickness. This finding has a direct regulatory implication: the welfare cost of mandatory batch clearing, such as that proposed under various periodic auction reforms, is not uniform across markets but is concentrated in the most liquid, high-frequency venues where the $T/2$ wait constitutes the largest foregone surplus.
\paragraph{The Theorem (ref) region is correctly identified.} The shaded band in the right panel corresponds to $\lambda \in [2.5, 7.5]$. Both welfare gaps are strictly positive within this band, confirming that the sufficient conditions of Theorem (ref) are binding in the right direction. Below $\lambda_{\min} \approx 2.5$, the $W^{\mathrm{LIT}} - W^{\mathrm{BATCH}}$ gap turns negative, indicating that the thin-market matching benefit of batch auctions temporarily outweighs their forced-waiting cost. This is the boundary case described in Section (ref): the sufficient condition $\lambda \geq \lambda_{\min}$ rules out precisely this regime.
Figure (ref) maps the welfare ranking across the full $(\lambda, \Delta)$ parameter space, sweeping $\lambda$ from 1 to 12 and $\Delta$ from 0.5 to 5. Each cell is color-coded according to which inequalities hold: green indicates the full ranking $W^{\mathrm{DARK}} > W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$ (Theorem 2), blue indicates only $W^{\mathrm{DARK}} > W^{\mathrm{LIT}}$, amber indicates only $W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$, and pink indicates that neither holds.
The heatmap reveals that the full ranking is highly robust to variation in valuation dispersion, $\Delta$. At all values from $\Delta = 0.5$ to $\Delta = 5$, the full ranking holds provided $\lambda$ is above a threshold that lies between approximately 2 and 4. The threshold shifts slightly upward as $\Delta$ increases: at $\Delta = 5$, the full ranking requires $\lambda \gtrsim 4$. This reflects the fact that higher valuation dispersion increases adverse selection in the dark pool, requiring a thicker market (higher $\lambda$) for high-cost traders to execute reliably via market orders. However, even at dispersion levels well beyond what is empirically plausible for a single liquid asset (say, $\Delta = 5$), the threshold remains below $\lambda = 4$, meaning the full ranking holds for the great majority of realistic market configurations.
The failure region is in the lower-left corner of the heatmap and is confined to combinations of very low arrival rates ($\lambda \lesssim 2$) and low-to-moderate dispersion ($\Delta \lesssim 4$). In this region, only the $W^{\mathrm{DARK}} > W^{\mathrm{LIT}}$ inequality survives; the batch auction's matching benefit outweighs its waiting cost. No cell exhibits the amber pattern ($W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$ without $W^{\mathrm{DARK}} > W^{\mathrm{LIT}}$), and the pink "no ranking" region is negligibly small and confined to a single corner cell. This strongly suggests that the dark pool's dominance over the lit exchange is the more robust of the two inequalities. Across the full parameter space explored, $W^{\mathrm{DARK}} \geq W^{\mathrm{LIT}}$ everywhere except at extreme combinations of high dispersion and very thin markets.
Figure (ref) plots $W^{\mathrm{BATCH}}(T)$ as a function of the batch interval $T \in [0.1, 5]$, alongside the $W^{\mathrm{LIT}}$ reference level.
The figure confirms Remark 4 visually: $W^{\mathrm{BATCH}}(T)$ is strictly decreasing in $T$ and converges to $W^{\mathrm{LIT}} \approx 0.250$ as $T \to 0$. $T = 1$ is the baseline value representing a one-second batch interval, comparable to the twelve-second interval used by Ethereum or the sub-second intervals of European Frequent Batch Auctions. $T = 0.5$ is an interval far shorter than any currently implemented periodic auction. The convergence is from above at very small $T$, reflecting the fact that at $T \approx 0.1$ the batch auction's matching benefit marginally exceeds its waiting cost, consistent with the thin-market regime identified in Section (ref). At $T=1$, $W^{\mathrm{BATCH}}$ has fallen to approximately 0.159, well below $W^{\mathrm{LIT}} \approx 0.250$. At $T = 5$ the welfare loss is severe: $W^{\mathrm{BATCH}} \approx 0.070$, roughly 72% below the lit exchange.
The slope of the welfare loss is steepest for small $T$ and flattens for large $T$. This convexity is consistent with the aggregate welfare-loss formula of Remark 1, $\Delta W_{\mathrm{wait}} = \mathbb{E}[C] \cdot (T/2) \cdot N$, combined with the endogenous reduction in $N$ as execution uncertainty discourages participation at longer intervals. The first effect is linear in $T$; the second amplifies it as marginal traders exit, producing the convex shape observed.
The practical implication is direct. Regulators contemplating periodic auction mandates face a quantitatively significant welfare trade-off: even at batch intervals as short as $T=0.5$, welfare is measurably below the continuous benchmark. The theoretical case for batch auctions as a welfare-improving reform, therefore, depends critically on whether the reduction in high-frequency arms races (not modeled here) outweighs the forced-waiting and execution-uncertainty costs quantified in this simulation.
Figure (ref) plots the fraction of traders who choose to participate in the market rather than take the outside option, as a function of $\lambda$ under each mechanism.
Three features stand out. First, the batch auction participation rate is uniformly the lowest of the three mechanisms and is strikingly flat across $\lambda$, remaining near 41% throughout the range $\lambda \in [1, 12]$. This flatness reflects the structure of the participation constraint under SIRO: execution uncertainty is determined by the fill rate $\bar{\rho}$, which depends on the ratio of buy and sell order flow rather than on the absolute arrival rate. As $\lambda$ increases, both sides of the market thicken proportionally, leaving $\bar{\rho}$ and therefore the participation incentive roughly unchanged. The 41% rate implies that approximately 59% of traders who would participate under a continuous mechanism are deterred by the combination of forced waiting and rationing risk, constituting a substantial deadweight loss.
Second, the dark pool and lit exchange participation rates are both increasing in $\lambda$ and converge toward each other as the market thickens, reaching approximately 49% at $\lambda = 12$. The dark pool maintains a small but consistent participation advantage over the lit exchange throughout the range: approximately 2--5 percentage points. This is consistent with the prediction that dark pool opacity reduces the cost of adverse selection for limit-order providers, attracting marginally more participation at the equilibrium cutoff $\bar{C}^*$. The convergence at high $\lambda$ reflects the diminishing marginal value of queue-state information in a thick, rapidly clearing book.
Third, and importantly, neither the dark pool nor the lit exchange approaches full participation (100%) even at $\lambda = 12$. The participation rate plateaus near 49%, limited by the participation constraint of Assumption 2. Traders with valuations close to $\hat{v}$ find that the expected gain from trade is insufficient to justify the fixed transaction cost $K$. This constraint is binding equally across mechanisms (since $K$ is common) and does not affect the welfare ranking, only the level.
Table (ref) collects the key quantitative findings of the simulation.
Taken together, the simulation results provide strong quantitative support for the theoretical welfare ranking of Theorem (ref). The ranking is not knife-edge: it holds across a wide region of the parameter space that comfortably encompasses plausible calibrations of real equity markets. The primary qualifications are the failure of the $W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$ inequality in very thin markets ($\lambda \lesssim 2.5$) and the declining magnitude of the dark-pool advantage as markets thicken. Both qualifications are consistent with the analytical structure of the model and do not overturn the main result in any empirically relevant regime.
The central mechanism driving $W^{\mathrm{DARK}} > W^{\mathrm{LIT}}$ in Theorem (ref) is that opacity eliminates strategic timing waste. The central cost of opacity is adverse selection: dark-pool traders cannot condition their limit prices on the observable book state $B_t$, so they may execute at prices that are less favourable relative to the fundamental value $\hat{v}$ than they would achieve in a transparent market. The proof of Theorem (ref) dismisses this cost as “second-order” on the grounds that prices are transfers. This section examines that dismissal carefully. We argue that the prices-are-transfers claim is correct within the model's assumptions but conceals a deeper and consequential trade-off: adverse selection does not harm total surplus directly through prices, but does so indirectly through its effect on participation. We characterise precisely when the participation effect is dominated by the timing-game saving, when it is not, and what this implies for the scope of the main result.
Adverse selection in dark pools operates through two distinct channels that the existing literature has not always kept separate.
\paragraph{Channel 1: The price channel.} When a limit-order trader in the dark pool posts at price $\bar{p}^*(C,s)$ without observing $B_t$, they may execute against a market order placed by a trader with superior information about $\hat{v}$. The resulting execution price is less favourable to the limit-order trader than the price they would have obtained in the lit exchange, where they could condition on $B_t$ and infer the likely informativeness of incoming market orders. This is the classical adverse selection described by OHara1998 and formalised in the dark-pool context by Zhu2013.
Within the welfare framework of this paper, the price channel generates no total surplus loss. Every cent lost by the limit-order trader on an adverse execution is gained by the market-order trader. Prices are pure transfers between the buyer and the seller; they affect the distribution of surplus but not its sum. Formally, for any matched pair $(i,j)$ with $s_i = +1$ (buyer) and $s_j = -1$ (seller), the joint surplus is $V_i - V_j$, which depends only on private valuations and is independent of the execution price $p$. The price channel is therefore irrelevant for aggregate welfare $W^M$ as defined in equation (61), and the dismissal in Theorem (ref) is correct with respect to this channel.
\paragraph{Channel 2: The participation channel.} The price channel affects only the division of surplus between matched counterparties. But adverse selection also affects whether traders participate at all. A limit-order trader in the dark pool who anticipates that their order will frequently execute against informed counterparties will earn a lower expected surplus per trade. If this expected surplus falls below their outside option, the trader exits the market entirely, and the gains from trade that would have been realised in a transparent market are destroyed. This is a genuine welfare loss: unlike a price transfer, a foregone trade eliminates surplus rather than redistributing it.
Formally, the participation condition for a dark-pool limit-order trader with type $(V, C, s)$ is (from equation 52 of the paper):
In the lit exchange, the corresponding condition is:
These two conditions are not identical. In the dark pool, the expected value $\bar{V}_{\mathrm{LO}}$ is computed over the unconditional distribution $\pi^{\mathrm{eq}}(B)$, which includes states with adverse book conditions. In the lit exchange, the trader can selectively participate when $B$ is favourable and exit when it is not. This option to condition participation on $B$ is valuable: it is a form of free entry into the advantageous states of the world. By eliminating this option, the dark pool reduces the expected surplus of some traders below their outside option, causing them to exit.
Let $\mathcal{X}^{\mathrm{DARK}}$ and $\mathcal{X}^{\mathrm{LIT}}$ denote the sets of types that participate under each mechanism. The argument above implies:
with strict containment for types near the participation margin $|V - \hat{v}| \approx \varepsilon(B)$. The welfare loss from the exclusion of types in $\mathcal{X}^{\mathrm{LIT}} \setminus \mathcal{X}^{\mathrm{DARK}}$ is:
where $\bar{B}$ is the mean book state. This integral is strictly positive whenever $\mathcal{X}^{\mathrm{LIT}} \setminus \mathcal{X}^{\mathrm{DARK}} \neq \varnothing$.
The net welfare advantage of the dark pool is:
The timing-game saving $\kappa(\lambda) > 0$ is pinned down by equation (ref) in the proof of Theorem (ref): it is bounded below by the product of the probability of thin book states, the expected excess waiting cost of high-cost traders, and the fraction of traders with $C > \bar{C}^*$. The participation loss $\Delta W_{\mathrm{participation}}$ depends on $\Delta$ (valuation dispersion) and on the severity of adverse selection in the dark pool.
We now characterise how $\Delta W_{\mathrm{participation}}$ depends on the model's parameters, and identify the conditions under which it dominates $\kappa(\lambda)$.
\paragraph{Dependence on $\Delta$.} Each trader near the participation margin has a gain from trade of approximately $|V - \hat{v}| - \varepsilon \approx 0$. When $\Delta$ is small, valuations are concentrated near $\hat{v}$ and many traders are near the participation margin; a small adverse-selection cost can push them out. When $\Delta$ is large, valuations are spread out and most traders are far from the margin; the adverse-selection cost must be very large to cause exit. Formally:
where $\eta(\Delta)$ is the fraction of the type space in the exclusion set. Crucially, $\eta(\Delta)$ itself depends on $\Delta$. As $\Delta \to 0$, the gains from trade vanish, so all traders are near the margin and $\eta(\Delta) \to 1$; as $\Delta \to \infty$, virtually no traders are near the margin and $\eta(\Delta) \to 0$. The product $2\Delta \cdot \eta(\Delta)$ is therefore non-monotone in $\Delta$: it is small for both very small $\Delta$ (small exclusion per trader) and very large $\Delta$ (small exclusion set), with a maximum at an intermediate level. The sufficient condition $\Delta < \Delta_{\max}$ in Theorem (ref) is precisely the requirement that $2\Delta \cdot \eta(\Delta) < \kappa(\lambda)$ for all $\lambda \in [\lambda_{\min}, \lambda_{\max}]$.
\paragraph{Dependence on $\sigma^2_v$.} The residual variance $\sigma^2_v$ of the fundamental value (Section 3.1) governs the severity of the price channel: higher $\sigma^2_v$ means incoming market orders are more often driven by private information about $v$, so the adverse execution price in the dark pool is further from the true value. This has no direct effect on total surplus (prices are transfers) but raises the adverse-selection discount that dark-pool limit-order traders must accept. A sufficiently high adverse-selection discount pushes some traders below their outside option, activating the participation channel. Our model abstracts from $\sigma^2_v$ in the equilibrium analysis by treating valuations as independent draws from $\mathrm{Uniform}[\hat{v}-\Delta, \hat{v}+\Delta]$. This is without loss of generality in a static model, but means that $\sigma^2_v$ enters only through $\Delta$ in the welfare comparison. In a richer model with asymmetric information about $\hat{v}$, the participation loss from adverse selection would be increasing in $\sigma^2_v$ independently of $\Delta$.
\paragraph{Dependence on $\gamma$.} The outside-option parameter $\gamma \in [0,1]$ is the fraction of private value realisable without market participation. When $\gamma$ is large, the outside option is attractive and many traders are near the participation margin regardless of market mechanism. A small adverse-selection cost in the dark pool then causes disproportionately large exit. Conversely, when $\gamma$ is small, the outside option is unattractive and traders remain in the dark pool even at adverse execution prices. The sufficient condition $K_o < K$ (Section 3.2) ensures the market attracts participation, but does not restrict $\gamma$ directly. For the welfare ranking to hold, we require that $\gamma$ is not so large that the dark pool's participation set $\mathcal{X}^{\mathrm{DARK}}$ is substantially smaller than $\mathcal{X}^{\mathrm{LIT}}$.
Our finding that the adverse selection cost is second-order under Theorem (ref)'s conditions stands in partial tension with results in two closely related papers.
Zhu2013 shows that dark pools attract disproportionately uninformed order flow, concentrating informed traders on the lit exchange. This is a venue-composition result: it predicts that the lit exchange becomes more adversely selected, not that total market-wide participation falls. The welfare implications of dark pools, Zhu notes, depend on elements outside the setting of his model — including how price information is used for production decisions, asset allocation, and capital formation. Within our framework, the Zhu mechanism is reflected in the distribution of types across mechanisms: if informed traders (high $|V - \hat{v}|$) prefer the lit exchange (because they can exploit observable book conditions) while uninformed traders (low $|V - \hat{v}|$) prefer the dark pool, then the dark pool is indeed the less adversely selected venue in terms of execution quality for a given participant. This reduces the adverse-selection cost we have been discussing — supporting, not undermining, the welfare advantage of the dark pool.
IyerJohariMoallemi2014 provide the most direct counterpoint. They find that the introduction of a dark pool can lead to greater transaction costs in the lit market and can decrease overall market welfare. Their mechanism is the participation channel: when adverse selection in the dark pool is severe enough, some intrinsic (value-motivated) traders are driven away from both venues, reducing total market participation and destroying gains from trade. Specifically, when the fundamental value is sufficiently different from the dark pool's transaction price, the combination of higher transaction costs in the open market and relatively high adverse selection costs in the dark pool drives some intrinsic traders away, leading to a net welfare loss. This is precisely $\Delta W_{\mathrm{participation}} > \kappa(\lambda)$ in the language of equation (ref).
The discrepancy between our result and IyerJohariMoallemi2014's is not a contradiction but a scope difference. Their model includes a competitive market maker whose spread endogenously responds to order flow composition; ours fixes the transaction-cost parameter $K$. In their setting, the migration of uninformed traders to the dark pool widens the lit-market spread, raising $K$ endogenously and amplifying the participation loss. In our static model with fixed $K$, this amplification mechanism is absent. The condition $\Delta < \Delta_{\max}$ effectively rules out the regime their model identifies as welfare-reducing: if $\Delta$ is small enough that adverse-selection costs are modest, the endogenous-spread amplification does not occur.
We now provide the formal characterisation of when the participation loss dominates, which the body of the paper left as a qualitative assertion.
Proposition (ref) has three direct regulatory implications that complement those of Section (ref).
\paragraph{Pre-trade transparency mandates.} Requiring dark pools to disclose their order book (converting them to lit exchanges) eliminates the information design that generates the timing-game saving $\kappa(\lambda)$ while having no effect on the participation loss $\Delta W_{\mathrm{participation}}$ (which is driven by adverse selection risk, not by pre-trade transparency per se). Transparency mandates therefore unambiguously reduce welfare when $\kappa(\lambda) > \bar\eta(\Delta)$ — precisely the condition under which the dark pool dominates.
\paragraph{Adverse selection disclosure.} An alternative regulatory approach would require dark pools to measure and disclose the adverse selection costs their participants face — for example, by reporting the average price concession on dark-pool executions relative to the lit midpoint (the “effective spread improvement” metric used by FINRA). Such disclosure does not eliminate the dark pool's information design (the book remains hidden) but allows participants to calibrate their participation decisions, mitigating the exclusion of marginal types. This would reduce $\Delta W_{\mathrm{participation}}$ without eliminating $\kappa(\lambda)$, thereby extending the parameter region in which the dark pool dominates.
\paragraph{Dark trading caps.} ESMA's Double Volume Cap (DVC) under MiFID II restricts the fraction of total order flow that can execute in dark venues. Within our framework, a dark trading cap limits the size of the dark-pool participation set $\mathcal{X}^{\mathrm{DARK}}$. If the cap binds, it excludes from the dark pool precisely those types nearest the participation margin — the traders for whom adverse selection is most costly relative to their gain from trade. This has the effect of reducing $\Delta W_{\mathrm{participation}}$ while also reducing the timing-game saving (fewer traders benefit from the dark pool's opacity). The net welfare effect of a cap depends on the curvature of the participation loss: if $\Delta W_{\mathrm{participation}}$ is convex in the size of the dark pool (marginal entrants have increasingly high adverse selection costs), a cap improves welfare. If it is concave, the cap is welfare-reducing.
The adverse selection trade-off has two faces. The price channel — the classical concern that dark-pool traders execute at unfavourable prices — is irrelevant for total surplus: adverse execution prices redistribute surplus between buyer and seller without destroying it. The participation channel — the concern that adverse selection drives marginal traders out of the market entirely — is the genuine welfare threat. Theorem (ref) establishes that the participation loss is dominated by the timing-game saving under $\Delta < \Delta_{\max}$ and $\lambda \in [\lambda_{\min}, \lambda_{\max}]$. Proposition (ref) characterises this condition precisely: the dark pool dominates if and only if $\kappa(\lambda) > \bar\eta(\Delta)$, and the threshold $\Delta^*(\lambda)$ is increasing in market thickness. This provides the formal grounding for the claim — asserted but not proved in the original paper — that adverse selection is second-order in the welfare comparison.
Theorem (ref) establishes the ranking $W^{\mathrm{DARK}} > W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$ under three sufficient conditions: a moderate arrival rate $\lambda \in [\lambda_{\min}, \lambda_{\max}]$, bounded valuation dispersion $\Delta < \Delta_{\max}$, and a short dark-pool reporting delay $\delta <T/2$. Any welfare comparison of this kind relies on modeling assumptions, and a responsible reading of the result requires understanding precisely which assumptions drive which inequalities, where the ranking breaks down, and what real-world conditions the sufficient conditions correspond to. This section addresses each of these questions in turn.
The two inequalities in the ranking have distinct analytical origins and, therefore, distinct load-bearing assumptions.
\paragraph{$W^{\mathrm{DARK}} > W^{\mathrm{LIT}}$: assumptions on information and strategic behavior.} This inequality rests on three assumptions working in concert. First, Assumption 3 (regularity of the arrival rate function) ensures that the lit-exchange equilibrium has a unique interior solution for the optimal limit price $p^*(C, s, B)$, characterized by the first-order condition (12). Without this regularity, traders' best responses to the observable book state could be non-unique or discontinuous, making the strategic timing cost ill-defined. Second, Assumption 4 (rational expectations in the dark pool) ensures that dark-pool traders form beliefs consistent with the equilibrium distribution of book states. Lemma 4 shows that under symmetric strategies these beliefs converge to the unconditional distribution $\pi^{\mathrm{eq}}(B)$, and Lemma 5 establishes that optimal strategies cannot condition on the unobservable $B$. Together, these imply that the strategic timing waste present in the lit exchange is eliminated in the dark pool. The strategic timing waste present in the lit exchange is formally captured by the difference between $\mathbb{E}[w^{\mathrm{LIT}} \mid C > \bar{C}^*]$ in equation (65) and $\mathbb{E}[w^{\mathrm{DARK}} \mid C > \bar{C}^*] = 0$ in equation (66). Third, the assumption that prices are transfers is required to conclude that the adverse selection created by opacity affects the distribution of surplus but not its total. This assumption is standard in the social-welfare tradition following OHara1998 and Zhu2013, but warrants scrutiny; we return to it in Section (ref).
\paragraph{$W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$: assumptions on waiting costs and execution certainty.} This inequality requires fewer structural assumptions and is, in that sense, more robust. The aggregate welfare loss from batch auctions has two components. The forced-waiting loss of Remark 1, $\Delta W_{\mathrm{wait}} = \mathbb{E}[C] \cdot (T/2) \cdot N$, follows directly from the SIRO matching rule and the batch structure: it holds for any positive $T$, any distribution of waiting costs with $\mathbb{E}[C] > 0$, and any positive measure of participants $N$. No regularity condition on $\lambda$ or $\Delta$ is required for this term to be strictly positive. The execution-uncertainty loss of Remark 2 requires only that the equilibrium fill rate $\bar{\rho} < 1$ for some set of traders of positive measure, which holds whenever order flow is subject to any stochastic imbalance. The imbalance is a near-universal feature of real markets. The only assumption that is genuinely load-bearing for $W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$ is Remark 4's implicit condition that $T > 0$ is not infinitesimally small. As $T \to 0$, the forced-waiting loss vanishes, and the batch auction converges to the lit exchange in welfare. The sufficient condition $\lambda \geq \lambda_{\min}$ ensures that the lit exchange provides sufficiently reliable immediate execution so that high-cost traders are not driven to prefer batch clearing as a coordination device.
Table (ref) catalogues the four boundary cases in which one or both inequalities fail, the economic mechanism behind each failure, and the parameter conditions under which each failure occurs.
Three observations follow from Table (ref). First, the failure of $W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$ is confined to very thin markets ($\lambda \lesssim 2.5$) or very short batch intervals ($T \lesssim 0.1$), neither of which characterizes the liquid equity venues for which this model is primarily relevant. Second, the failure of $W^{\mathrm{DARK}} > W^{\mathrm{LIT}}$ requires either extreme valuation dispersion or a breakdown of the "prices are transfers" assumption. The former does not occur within the empirically plausible parameter range explored in Section (ref); the latter is an assumption of the model that should be relaxed in future work. Third, the heatmap of Figure (ref) shows that no cell in the explored $(\lambda, \Delta)$ grid exhibits the pattern $W^{\mathrm{LIT}} > W^{\mathrm{BATCH}}$ without $W^{\mathrm{DARK}} > W^{\mathrm{LIT}}$, suggesting that the dark-pool dominance over the lit exchange is the more robust of the two inequalities.
We now examine each of the five modeling assumptions in turn and assess the direction and likely magnitude of the bias introduced if it is relaxed.
\paragraph{Assumption 1: Uniform valuations and exponential waiting costs.} The Uniform$[\hat{v} - \Delta, \hat{v} + \Delta]$ distribution for private valuations ensures a uniform density of gains from trade, which simplifies the welfare integrals. The key property used in the proof of Theorem (ref) is that the participation cutoff $\varepsilon(B)$ is well-defined and continuous in $B$. This holds for any continuously differentiable valuation density with bounded support. The Uniform distribution is therefore not special; any log-concave density on a compact interval yields the same qualitative results.
The Exponential distribution for waiting costs $C$ ensures that the cutoff $C^*(V, B)$ separating market and limit orders is finite, and that the mean waiting cost $\mathbb{E}[C] = 1/\mu_C$ is well-defined. Any distribution with a finite first moment and a monotone hazard rate preserves the monotone cutoff property of Lemma 2. Fat-tailed distributions (e.g., Pareto waiting costs) would increase the aggregate waiting cost in the lit exchange disproportionately, strengthening $W^{\mathrm{DARK}} > W^{\mathrm{LIT}}$, since high-cost traders are more numerous and their timing-game waste is more severe.
\paragraph{Assumption 2: Participation constraint.} Assumption 2 requires that expected gains from trade exceed fixed costs plus expected waiting costs: $\mathbb{E}[|V_i - \hat{v}|] > K + \mathbb{E}[C_i] \cdot \mathbb{E}[w_i]$. This ensures strictly positive market participation. If this constraint is violated, the market unravels and welfare is zero under all three mechanisms. The constraint is most likely to bind in thin markets (high $\mathbb{E}[w_i]$) with large fixed costs ($K$ close to $\Delta$). Within the parameter range explored in Section (ref), the constraint holds with a comfortable margin at all grid points.
\paragraph{Assumption 3: Regularity of the arrival rate function.} This assumption, which ensures a unique interior solution for the optimal limit price in the lit exchange (Lemma 1), is the most technically restrictive. Its key content is that $\lambda(p, B, s)$ is twice continuously differentiable and concave in $p$, so that the first-order condition (12) has a unique solution. This is satisfied by the normal-CDF parametrization used in the simulation ($\lambda \propto \Phi(\cdot)$) and by any log-concave arrival rate. The arrival rate does not have to be log-concave: for example, if the order book has discrete price levels creating a step function in $\lambda$. If the arrival rate is not log-concave, however, the optimal limit price may not be unique, and the equilibrium characterization of Proposition 1 requires modification. The qualitative result that observable book states create strategic complementarities in submission timing does not depend on uniqueness and is likely to survive in models with discrete price grids, though the welfare calculation requires numerical methods.
\paragraph{Assumption 4: Rational expectations in the dark pool.} The requirement that dark-pool traders form beliefs consistent with the equilibrium distribution $\pi^{\mathrm{eq}}(B)$ is standard in Bayesian equilibrium analysis. The traders may have "incorrect" beliefs. For example, they may use a misspecified model of order flow. In this case, the dark-pool equilibrium of Proposition 2 need not obtain. In particular, if traders systematically underestimate adverse selection in the dark pool, they may over-participate, reducing their realized welfare below the rational-expectations prediction. The direction of this bias is ambiguous: under- estimation of adverse selection raises participation (moving welfare toward $W^{\mathrm{LIT}}$ from below) but also raises realized losses (moving welfare downward). The net effect on the welfare ranking depends on parameters.
\paragraph{Assumption 5: Stationary batch characteristics.} The batch auction analysis assumes that clearing prices and fill rates are stationary across batches: $\mathbb{E}[p_k] = \bar{p}$ and $\rho_k = \bar\rho$ for all $k$ (Assumption 5 in the Appendix). This is a steady-state approximation. In practice, batch characteristics vary with market conditions. Thick liquidity periods produce both more orders and higher $\rho_k$, generating a positive correlation between fill rates and market activity. During thick periods, the welfare loss from execution uncertainty may be lower than predicted because the periods in which rationing occurs are also periods in which the opportunity cost of non-execution is lower. The direction of this correlation suggests that Assumption 5 leads to a slight overstatement of the batch auction's welfare cost relative to a fully dynamic model.
The model is most directly applicable to settings that satisfy the following four conditions.
It is instructive to compare our welfare ranking with prior results in the literature to understand what the present model adds and where it agrees or disagrees.
Leshno2022 shows that in a single-sided waiting list with overloaded demand, SIRO dominates FCFS in ex-ante welfare because SIRO discourages socially excessive effort to secure priority. Applied na\"{i}vely, this suggests batch auctions (SIRO) should dominate lit exchanges (FCFS). Our result reverses this in the two-sided market setting because the lit exchange allows high-cost traders to execute immediately via market orders, bypassing the queue entirely. This option to use a market order is absent in the single-sided waiting list of Leshno2022. However, this option fundamentally changes the welfare calculation.
CheTercieux2023 show that uninformed FCFS (FCFS without knowledge of queue position) dominates SIRO when agents are uninformed of their position. This directly supports $W^{\mathrm{DARK}} > W^{\mathrm{BATCH}}$. Our contribution is to establish $W^{\mathrm{DARK}} > W^{\mathrm{LIT}}$ as an intermediate result in a financial market with endogenous participation, adverse selection, and an outside option, and to characterize the parameter region in which all three comparisons hold simultaneously.
Zhu2013 shows that dark pools attract more uninformed order flow, which improves price discovery in lit exchanges but concentrates adverse selection in the dark pool. Our model is consistent with this finding but approaches it from a different angle: we take the information structure as given and ask how it affects the total surplus generated by the matching process. The adverse selection documented by Zhu2013 affects the distribution of surplus between informed and uninformed traders. However, in our model, the adverse selection does not affect the total surplus (because prices are transfers). The tension between these two conclusions is resolved by the scope condition: when adverse selection is so severe that participation is materially lower in the dark pool than in the lit exchange, the "prices are transfers" assumption breaks down and the welfare advantage of opacity may disappear. The magnitude of this participation effect is ultimately an empirical question that our theoretical framework cannot answer alone.
Our model analyzes a single trading period. In reality, the same asset trades repeatedly, creating dynamic linkages. We briefly discuss two key dynamic effects.
Our theoretical findings suggest several policy considerations:
We have developed a formal game-theoretic model comparing three major market mechanisms: lit exchanges (FCFS continuous auctions), dark pools (FCFS with hidden books), and batch auctions (SIRO). Our main theoretical contribution is Theorem 1, which shows that under moderate arrival rates and bounded adverse selection, dark pools dominate both alternatives in ex-ante welfare. The key insight is that information opacity eliminates socially wasteful strategic behavior, specifically, costly timing games that arise when traders can observe order book depth and position themselves optimally. While dark pools create adverse selection (prices are less informative), this affects wealth distribution, not total surplus, in our model.
Our welfare ranking $W^{DARK} > W^{LIT} > W^{BATCH}$ holds under specific conditions. These theoretical predictions align with observed market structure: the coexistence of many dark pools alongside fewer lit exchanges and one batch venue suggests efficient sorting.
By providing rigorous foundations for comparing market mechanisms, this paper contributes to the literature on market microstructure, mechanism design, and the economics of information. The framework can be extended to analyze other trading venues (e.g., decentralized exchanges in cryptocurrency markets) and inform ongoing regulatory debates about market structure.